# To express: Kepler’s Third Law of planetary motion as equation.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.11, Problem 42E

(a)

To determine

## To express: Kepler’s Third Law of planetary motion as equation.

Expert Solution

The equation of Kepler’s third law of planetary motion is T2=kd3 .

### Explanation of Solution

Kepler’s third law of planetary motion:

The square of the period (T) of a planet is directly proportional to the cube of its average distance (d) from the sun.

Direct variation:

If the quantities a and b are related by the equation a=Cb then for C0 , a is directly proportional to b and C is called constant of proportionality.

T2 is proportional to d3

Mathematically,

T2=kd3

Here,

T is the time planet take to complete one revolution around the sun, d is average distance of planet from the sun.

k is constant of proportionality.

Thus Kepler’s third law as equation is T2=kd3 .

(b)

To determine

### To find: The constant of proportionality (k) .

Expert Solution

The value of constant of proportionality is 1.65×1019 .

### Explanation of Solution

Given:

Time period for our planet is 365days .

Average distance between sun and our planet is 93×106miles .

Calculation:

Kepler’s Law as equation,

T2=kd3 (1)

Substitute 365 for T and 93×106 for d in equation (1) and solve for k .

3652=k(93×106)3133225=k(8.04×1023)1332258.04×1023=k1.65×1019=k

Thus the value of constant of proportionality is 1.65×1019 .

(c)

To determine

### To find: The time period of Neptune.

Expert Solution

The time period of Neptune is 58387days .

### Explanation of Solution

Given:

Distance of Neptune from Sun is 2.79×109mi .

Calculation:

The value of constant of proportionality from part(b) is 1.65×1019 .

Use Kepler’s third Law,

T2=kd3

Substitute 2.79×109 for d and 1.65×1019 for k in equation (1) and solve for T .

T2=(1.65×1019)(2.79×109)3T2=(1.65×1019)(2.17×1028)T2=35805×105

Take square root of above equation and get the value of T .

T=58387.27

Thus, time period of Neptune is approximately 58387days .

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